MODELS OF G-SPECTRA AS PRESHEAVES OF SPECTRA

Transcription

1 MODELS OF G-SPECTRA AS PRESHEAVES OF SPECTRA BERTRAND GUILLOU AND J.P. MAY Abstract. Let G be a finite group. We give Quillen equivalent models for the category of G-spectra as categories of spectrally enriched functors from explicitly described domain categories to nonequivariant spectra. Our preferred model is based on equivariant infinite loop space theory applied to elementary categorical data. It recasts equivariant stable homotopy theory in terms of point-set level categories of G-spans and nonequivariant spectra. We also give a more topologically grounded model based on equivariant Atiyah duality. Contents Introduction 2 1. The S -category GA and the S G -category A G The bicategory GE of G-spans The precise statement of the main theorem The G-bicategory E G of spans: intuitive definition The G-bicategory E G of spans: working definition The categorical duality maps The proof of the main theorem The equivariant approach to Theorem Results from equivariant infinite loop space theory The self-duality of Σ G (A +) The proof that A G is equivalent to D G Identifications of suspension G-spectra and of tensors with spectra Atiyah duality for finite G-sets The categories GZ, GD, and D G Space level Atiyah duality for finite G-sets The weakly unital categories GB and B G The category of presheaves with domain GB Appendix: Enriched model categories of G-spectra Presheaf models for categories of G-spectra Comparison of presheaf models of G-spectra Suspension spectra and fibrant replacement functors in GS Suspension spectra and smash products in GZ Appendix: Whiskering GE to obtain strict unit 1-cells 36 References 37 Date: July 7,

2 2 BERTRAND GUILLOU AND J.P. MAY Introduction The equivariant stable homotopy category is of fundamental importance in algebraic topology. It is the natural home in which to study equivariant stable homotopy theory, a subject that has powerful and unexpected nonequivariant applications. For recent examples, it plays a central role in the solution of the Kervaire invariant problem by Hill, Hopkins, and Ravenel, it is central to calculations of topological cyclic homology and therefore to calculations in algebraic K-theory made by Angeltveit, Gerhardt, Hesselholt, Lindenstrauss, Madsen, and others, and it plays an interesting role by analogy and comparision in the work of Voevodsky and others in motivic stable homotopy theory. It is also of great intrinsic interest. Setting up the equivariant stable homotopy category with its attendant model structures takes a fair amount of work. The first version was due to Lewis and May [20] and more modern versions that we shall start from are given in Mandell and May [23]. A result of Schwede and Shipley [36], reproven in [8], asserts that any stable model category M is equivalent to a category Pre(D, S ) of spectrally enriched presheaves with values in a chosen category S of spectra. However, the domain S -category D is a full S -subcategory of M and typically is as inexplicit and mysterious as M itself. From the point of view of applications and calculations, this is therefore only a starting point. One wants a more concrete understanding of the category D. We shall give explicit equivalents to the domain category D in the case when M = GS is the category of G-spectra for a finite group G, and we fix a finite group G throughout. We shall define an S -category (or spectral category) GA by applying a suitable infinite loop space machine to simply defined categories of finite G-sets. The spectral category GA is a spectrally enriched version of the Burnside category of G. We shall prove the following result. Theorem 0.1 (Main theorem). There is a zig-zag of Quillen equivalences GS Pre(GA, S ) relating the category of G-spectra to the category of spectrally enriched contravariant functors GA S. As usual, we call such functors presheaves. We reemphasize the simplicity of our spectral category GA : no prior knowledge of G-spectra is required to define it. We give a precise description of the relevant categorical input and restate the main theorem more precisely in 1. The central point of the proof is to use equivariant infinite loop space theory to construct the spectral category GA from elementary categories of finite G-sets. We prove our main theorem in 2, using the equivariant Barratt-Priddy-Quillen (BPQ) theorem to compare GA to the spectral category GD given by the suspension G-spectra Σ G (A +) of based finite G-sets A +, which is a standard choice for application of the theorem of Schwede and Shipley to GS. The classical Burnside category of isomorphism classes of spans of finite G-sets leads to a calculation of the homotopy category HoGD (see Theorem 1.11 below), and GA starts from the bicategory of such spans, in which isomorphisms of spans give the 2-cells. Intuitively, Mackey functors can be viewed as functors from HoGD to abelian groups, and the result of Schwede and Shipley says that G-spectra can be viewed as functors from GD to spectra. We are lifting the standard purely algebraic

3 MODELS OF G-SPECTRA AS PRESHEAVES OF SPECTRA 3 understanding of Mackey functors to obtain an analogous algebraic understanding of G-spectra as functors from GA to spectra. Thus the slogan is that G-spectra are spectral Mackey functors. It is crucial to our work that the G-spectra Σ G (A +) are self-dual. Our original proof took this as a special case of equivariant Atiyah duality ( 3.2), thinking of A as a trivial example of a smooth closed G-manifold. We later found a direct categorical proof ( 2.3) of this duality based on equivariant infinite loop space theory and the equivariant BPQ theorem. This allows us to give an illuminating new proof of the required self-duality as we go along. We give an alternative model for the category of G-spectra in terms of classical Atiyah duality in 3. An appendix, 4 provides some background on the two model categories of G-spectra used here, equivariant orthogonal spectra and equivariant S-modules, and describes and compares the specialization of [8] to those categories that provides the starting point for our work. We take what we need from equivariant infinite loop space theory as a black box in this paper. The additive and multiplicative space level theories are worked out in [32] and [11], respectively. The generalization from space level to category level input is based on general (and not necessarily equivariant) categorical coherence theory that is worked out in [12, 13, 14]. What is needed for this paper is a small part of the full story and is put together in a relatively short companion paper [15]. We thank a diligent referee for demanding a reorganization of our original paper. We also thank Angelica Osorno and Inna Zakharevich for very helpful comments, and we especially thank Osorno and Anna Marie Bohmann for catching an error in the handling of pairings in earlier versions of this work. That error is one reason for the very long delay in the publication of this paper, which was first posted on ArXiv several years ago, on August 21, The delay is no fault of this journal. In the interim, we teamed with Osorno and Mona Merling to fully work out the relevant infinite loop space theory, which turned out to be both surprisingly demanding and unexpectedly interesting. Also in the interim, Bohmann and Osorno [2] made concrete applications of this paper for the construction of genuine G- spectra from categorical input data. A small error 1 in their paper is corrected in the short appendix, 5, of this paper. Further applications to the concrete construction of genuine G-spectra are in development in their work and in work of Cary Malkievich and Merling [22]. We also note that Clark Barwick [1], inspired by our work, has given an abstract infinity categorical variant of our main result. During the delay, Jonathan Rubin combed through our draft and caught a great many errors of detail and infelicities. Needless to say, we are responsible for all that remain. This work was partially supported by Simons Collaboration Grant No held by the first author. 1. The S -category GA and the S G -category A G In this paper, S denotes the category of (nonequivariant) orthogonal spectra. See 4 for some discussion of the comparison between models of G-spectra. We first define the S -category GA and restate our main theorem. We shall avoid categorical apparatus, but conceptually GA can be viewed as obtained by applying a nonequivariant infinite loop space machine K to a category GE enriched in 1 We are grateful to Angelica Osorno for helping us discover and fix this error.

4 4 BERTRAND GUILLOU AND J.P. MAY permutative categories. 2 The term in quotes can be made categorically precise [7, 16, 35], but we shall use it just as an informal slogan since no real categorical background is necessary to our work here: we shall give direct elementary definitions of the examples we use, and they do satisfy the axioms specified in the cited sources. We then define a G-category 3 E G enriched in permutative G-categories, from which GE is obtain by passage to G-fixed subcategories. Finally, we outline the proof of the main theorem, which is obtained by applying an equivariant infinite loop space machine K G to E G The bicategory GE of G-spans. In any category C with pullbacks, the bicategory of spans in C has 0-cells the objects of C. The 1-cells and 2-cells A B are the diagrams (1.1) B D A and Composites of 1-cells are given by (chosen) pullbacks (1.2) F E D D B = A. E. C B A. The identity 1-cells are the diagrams A = A = A. The associativity and unit constraints are determined by the universal property of pullbacks. Observe that the 1-cells A B can just as well be viewed as objects over B A. Viewed this way, the identity 1-cells are given by the diagonal maps A A A. Our starting point is the bicategory of spans of finite G-sets. Here the disjoint union of G-sets over B A gives us a symmetric monoidal structure on the category of 1-cells and 2-cells A B for each pair (A, B). We can think of the bicategory of spans as a category enriched in the category of symmetric monoidal categories. Again, the notion in quotes does not make obvious mathematical sense since there is no obvious monoidal structure on the category of symmetric monoidal categories, but category theory due to the first author [7] (see also [16, 35]) explains what these objects are and how to rigidify them to categories enriched in permutative categories. We repeat that we have no need to go into such categorical detail. Rather than apply such category theory, we give a direct elementary construction of a strict structure that is equivalent to the intuitive notion of the category enriched in symmetric monoidal categories of spans of finite G-sets. We first define a bipermutative category GE (1) that is equivalent to the symmmetric bimonoidal category of finite G-sets. Definition 1.3. Any finite G-set is isomorphic to one of the form A = (n, α), where n = {1,, n}, α is a homomorphism G Σ n, and G acts on n by 2 A permutative category is a symmetric strict monoidal category. 3 In general, we understand a G-category to be a category internal and not just enriched in G-sets, meaning that G can act on both objects and morphisms.

5 MODELS OF G-SPECTRA AS PRESHEAVES OF SPECTRA 5 g i = α(g)(i) for 1 i n. We understand finite G-sets to be of this restricted form from now on. A G-map f : (m, α) (n, β) is a function f : m n such that f α(g) = β(g) f for g G. The morphisms of GE (1) are the isomorphisms (n, α) (n, β) of G-sets. The disjoint union D E of finite G-sets D = (s, σ) and E = (t, τ) is (s + t, σ+τ), with σ + τ being the evident block sum G Σ s+t. With the evident commutativity isomorphism, this gives the permutative category GE (1) of finite G-sets; the empty finite G-set is the unit for. To define the cartesian product, for each s and t let λ s,t : st s t denote the lexicographic ordering. Then D E is (st, σ τ) where σ τ is the permutation st λs,t s t σ τ s t λ 1 s,t st. There is again an evident commutativity isomorphism, and and give GE ( ) a structure of bipermutative category in the sense of [30]; the multiplicative unit is the trivial G-set 1 = (1, ε), where ε(g) = 1 for g G. As we will need it later, we also introduce the reordering permutation τ s,t Σ st, defined as the composition st λs,t s t = t s λ 1 t,s ts = st. We may view GE (1) as the category of finite G-sets over the one point G-set 1, and we generalize the definition as follows. Definition 1.4. For a finite G-set A, we define a permutative category GE (A) of finite G-sets over A. The objects of GE (A) are the G-maps p: D A. The morphisms p q, q : E A, are the G-isomorphisms f : D E such that q f = p. Disjoint union of G-sets over A gives GE (A) a structure of permutative category; its unit is the empty set over A. When A = 1, GE (A) is the ( additive ) permutative category of the previous definition. Remark 1.5. There is also a product : GE (A) GE (B) GE (A B). It takes (D, E) to D E, where D and E are finite G-sets over A and B, respectively. This product is also strictly associative and unital, with unit the unit of GE (1), and it has an evident commutativity isomorphism. Restriction to the object 1 gives the multiplicative permutative category of Definition 1.3. This product distributes over and almost makes the enriched category GE of the next definition into a category enriched in permutative categories, in the sense defined in [7]. There is no obvious sense since the category of permutative categories is not monoidal. The almost refers to the fact that the category we define does not have a strict unit, a problem that was encountered in [2] and is fixed in 5 below. Definition 1.6. We define a bicategory GE with a permutative category of hom objects for each pair of objects as follows. The 0-cells of GE are the finite G- sets, which may be thought of as the categories GE (A). The permutative category GE (A, B) of 1-cells and 2-cells A B is GE (B A), as defined in Definition 1.4. The 1-cells are thought of as spans and the 2-cells as isomorphisms of spans. The composition : GE (B, C) GE (A, B) GE (A, C) is defined via pullbacks, as in the diagram (1.2). The diagonal map A : A A A serves as a unit 1-cell. Precisely, following [2, 7.2], we choose the pullback F

6 6 BERTRAND GUILLOU AND J.P. MAY in (1.2) to be the sub G-set of E D, ordered lexicographically, consisting of the elements (e, d) such that d and e map to the same element of B. Remark 1.7. This bicategory is almost a 2-category. The composition of spans is strictly associative, but if A 2 then A : A A A acts as a strict unit only on the right and so should be called a pseudo-unit 1-cell. The point is that with our chosen model for the pullback, the left map in the span composition B p 1 B E f p 2 E B B A must be order-preserving. Therefore, if f is not order-preserving, then B E E. However, in view of the evident commutative diagram B p 1 f B E E p 2 g g p 2 A, the function p 2 specifies a reordering isomorphism of spans (1.8) B E l B,E E In 5, we show how to whisker the pseudo-unit 1-cells to obtain an equivalent construction GE that still has a strictly associative composition but now has strict two-sided unit 1-cells. The construction is closely analogous to the usual whiskering of a degenerate basepoint in a space to obtain a nondegenerate basepoint. While we give precise details where needed, replacing GE by GE is a minor quibble. Remark 1.9. We are suppressing some categorical details that are irrelevant to our work. The composition distributes over coproducts, and it should be defined on a tensor product rather than a cartesian product of permutative categories. Such a tensor product does in fact exist [16], but we shall not use the relevant category theory. Rather we will change notation to since the composition is a pairing that gives rise to a pairing defined on the smash product of the spectra constructed from GE (B, C) and GE (A, B). The passage from pairings of permutative categories to pairings of spectra has a checkered history even nonequivariantly, 4 and it is here that a mistake occurred in earlier versions of this paper. As explained in [15], categorical strictification and the full development of multiplicative equivariant infinite loop space theory resolve the relevant issues. Remark It is helpful to observe that the composition just defined can be viewed as a composite of maps of finite G-sets induced contravariantly and covariantly by the maps of finite G-sets id id C B B A π C B A C A, where π : C B A C A is the projection. g 4 That starts from [27], which is modernized, corrected, and generalized in [15].

7 MODELS OF G-SPECTRA AS PRESHEAVES OF SPECTRA 7 Before beginning work, we recall an old result that motivated this paper. The category [GE ] of G-spans is obtained from the bicategory GE of G-spans by identifying spans from A to B if there is an isomorphism between them. Composition is again by pullbacks. We add spans from A to B by taking disjoint unions, and that gives the morphism set [GE ](A, B) a structure of abelian monoid. We apply the Grothendieck construction to obtain an abelian group of morphisms A B. This gives an additive category Ab[GE ]. The following result is [20, V.9.6]. Let HoGD denote the full subcategory of the homotopy category HoGS of G-spectra whose objects are the G-spectra Σ G (A +), where A runs over the finite G-sets. Theorem The categories HoGD and Ab[GE ] are isomorphic The precise statement of the main theorem. Infinite loop space theory associates a spectrum KA to a permutative category A. There are several machines available and all are equivalent [29]. Since it is especially convenient for the equivariant generalization, we require K to take values in orthogonal spectra [24], but symmetric spectra would also work. As in the axiomatization of [29], we require K to take values in Ω-spectra and we require a natural group completion η : BA (KA ) 0. The objects a A are the vertices of the nerve of A and are thus points of BA hence, via η, points of (KA ) 0. Therefore each a determines a map S KA, where S is the sphere spectrum. Since S is closed symmetric monoidal under the smash product, it makes sense to enrich categories in S. Our preferred version of spectral categories is categories enriched in S, abbreviated S -categories. Model theoretically, S is a particularly nice enriching category since its unit S is cofibrant in the stable model structure and S satisfies the monoid axiom [24, 12.5]. When a spectral category D is used as the domain category of a presheaf category, the objects and maps of the underlying category are unimportant. The important data are the morphism spectra D(A, B), the unit maps S D(A, A), and the composition maps D(B, C) D(A, B) D(A, C). The presheaves D op S can be thought of as (right) D-modules. Definition We define a spectral category GA. Its objects are the finite G-sets A, which may be viewed as the spectra KE (A). Its morphism spectra GA (A, B) are the spectra KGE (A, B). Its unit maps S GA (A, A) are induced by the points I A GE (A, A) and its composition is induced by composition in GE. GA (B, C) GA (A, B) GA (A, C) As written, the definition makes little sense: to make the word induced meaningful requires properties of the infinite loop space machine K that we will spell out in 2.2. Once this is done, we will have the presheaf category Pre(GA, S ) of S -functors (GA ) op S and and S -natural transformations. As shown for example in [8], it is a cofibrantly generated model category enriched in S, or an S -model category for short. As shown in [23], the category GS of (genuine) orthogonal G-spectra is also an S -model category. Our main theorem can be restated as follows. Theorem 1.13 (Main theorem). There is a zigzag of enriched Quillen equivalences connecting the S -model categories GS and Pre(GA, S ).

8 8 BERTRAND GUILLOU AND J.P. MAY Therefore G-spectra can be thought of as constructed from the very elementary category GE enriched in permutative categories, ordinary nonequivariant spectra, and the black box of infinite loop space theory. The following reassuring result falls out of the proof. Let Orb denote the orbit category of G. For a G-spectrum X, passage to H-fixed point spectra for H G defines a functor X : Orb op S. Analogously, a presheaf Y Pre(GA, S ) restricts to a functor Orb op S. Corollary The zigzag of equivalences induces a natural zigzag of equivalences between the fixed point orbit functor on G-spectra and the restriction to orbits of presheaves. Thus, if X is a fibrant G-spectrum that corresponds to the presheaf Y, then X H is equivalent to Y (G/H). Remark For any n, the homotopy groups π n (X H ) define a Mackey functor, and so do the homotopy groups π n (Y (G/H)). The corollary implies an isomorphism between these Mackey functors. Remark There are several missing ingredients needed for a fully satisfactory theory. To avoid undue length, we will not prove the analogue of Corollary 1.14 for geometric fixed points and we shall not treat change of group functors. We do not believe there are any essential difficulties. However, more importantly, we have not described the behavior of smash products under the equivalences of Theorem This problem deserves study both in our work and in related work of others. The category Pre(GA, S ) is symmetric monoidal (under Day convolution). The obvious guess is that the zigzag connecting it to GS is a zigzag of symmetric monoidal Quillen equivalences. We see how the problem can be attacked, but we believe that the obvious guess is wrong. We intend to return to this question elsewhere. Remark Much of what we do applies to G-spectra indexed on an incomplete universe, although we have not thought through full details. We must then restrict attention to those finite G-sets A that embed in the given universe, so that Atiyah duality applies to the orbit G-spectra Σ G (A +). By [19], duality fails for orbits that do not embed in the universe. To mesh with the notion of generators for a stable model category, the weak equivalences must then be defined in terms of the homotopy groups of H-fixed point spectra for those H such that G/H embeds in the given universe. Corollary 1.14 would have to be restricted similarly The G-bicategory E G of spans: intuitive definition. Everything we do depends on first working equivariantly and then passing to fixed points. We fix some generic notations. For a category C, let GC be the category of G-objects in C and G-maps between them. Let C G be the G-category of G-objects and nonequivariant maps, with G acting on morphisms by conjugation. The two categories are related conceptually by GC = (C G ) G. The objects, being G-objects, are already G-fixed; we apply the G-fixed point functor to hom sets. More generally, we can start with a category C with actions by G on its objects and again define a category GC of G-maps and a G-category C G with G-fixed category GC. The reader may prefer to think of GC as a category enriched in G-categories, with enriched hom objects the G-categories C G (A, B) for G-objects A and B. We apply this framework to the category of finite G-sets. We have already defined the G-fixed bicategory GE, and we shall give two definitions of G-bicategories E G

9 MODELS OF G-SPECTRA AS PRESHEAVES OF SPECTRA 9 with fixed point bicategories equivalent to GE. The first, given in this section, is more intuitive, but the second is more convenient for the proof of our main theorem. Let U be a countable G-set that contains all orbit types G/H infinitely many times. Again let A, B, and C denote finite G-sets, but now think of the D, E, and F of (1.1) and (1.2) as finite subsets of the G-set U; these subsets need not be G-subsets. The action of G on U gives rise to an action of G on the finite subsets of U: for a finite subset D of U and g G, gd is another finite subset of U. Definition We define a G-category EG U (A). The objects of E G U (A) are the nonequivariant maps p: D A, where A is a finite G-set and D is a finite subset of U. The morphisms f : p q, q : E A, are the bijections f : D E such that q f = p. The group G acts on objects and morphisms by sending D to gd and sending a bijection f : D E over A to the bijection gf : gd ge over A given by (gf)(gd) = g(f(d)). Definition We define a bicategory EG U with objects the finite G-sets and with G-categories of morphisms between objects given by EG U (A, B) = E G U (B A). Thinking of the objects of EG U (A, B) as nonequivariant spans B D A, composition and units are defined as in Definition 1.6. Observe that taking disjoint unions of finite sets over A will not keep us in U and is thus not well-defined. Therefore the EG U (A) are not even symmetric monoidal (let alone permutative) G-categories in the naive sense of symmetric monoidal categories with G acting compatibly on all data The G-bicategory E G of spans: working definition. We shall work with a less intuitive definition of E G, one that solves the problem of disjoint unions by avoiding any explicit use of them. It uses an especially convenient E operad of G-categories, denoted P G. We recall it from [9], where we define a genuine permutative G-category to be an algebra over P G. More generally, in [12] we define a genuine symmetric monoidal G-category to be a pseudoalgebra over P G, but we will not need that notion here. Such pseudoalgebras provide input for an equivariant infinite loop space machine. To define P G, we apply our general point of view on equivariant categories to the category Cat of small categories. Thus, for G-categories A and B, let Cat G (A, B) be the G-category of functors A B and natural transformations, with G acting by conjugation, and let GCat(A, B) = Cat G (A, B) G be the category of G-functors and G-natural transformations. Definition Let EG be the groupoid 5 with object set G and a unique morphism, denoted (h, k), from k to h for each pair of objects. Let G act from the right on EG by h g = hg on objects and (h, k) g = (hg, kg) on morphisms. The objects of E G are the finite G-sets A = (n, α), regarded as discrete (identity morphisms only) G-categories. Define P(j) = EΣ j ; this is the jth category of an E operad of categories whose algebras are the permutative categories [28]. Define P G (j) to be the G-category Cat G (EG, EΣ j ) = Cat G (EG, P(j)). 5 While EG is isomorphic as a G-category to the translation category of G, the action of G on that category is defined differently, as is explained in [10, Lemma 1.7]. Our EG is the chaotic category of G, often denoted G.

10 10 BERTRAND GUILLOU AND J.P. MAY Here G acts trivially on EΣ j. The left action of G on P G (j) is induced by the right action of G on EG, and the right action of Σ j is induced by the right action of Σ j on EΣ j. The functor Cat G (EG, ) is product preserving and the operad structure maps are induced from those of P. We interpret P(0) and P G (0) to be trivial categories; P G (1) is also trivial, with unique object denoted id. Definition Regard a finite G-set A as a discrete G-category (identity morphisms only). Define the G-category E G (A) by (1.22) E G (A) = n 0 P G (n) Σn A n = ( n 1 P G (n) Σn A n ) +. We interpret the term with n = 0 to be a trivial base category, which explains the second equality, and we identify the term with n = 1 with A. The following result is neither obvious nor difficult. It is proven in [9, Theorem 5.5], where it is one ingredient in a categorical proof of the tom Dieck splitting theorem. Theorem The G-fixed permutative category E G (A) G is naturally isomorphic to the permutative category GE (A) of Definition 1.4. The starting point of the proof is the observation that a functor EG EΣ n is uniquely determined by its object function G Σ n. In particular, for a finite G- set B = (n, β) we may view the group homomorphism β : G Σ n as an object of the category P G (n). With a little care, we see that a G-fixed object (β; a 1,, a n ) of P G (n) Σn A n can be interpreted as a G-map B A and that all finite G-sets over A are of this form. Remark Conceptually, Definition 1.21 hides an important identification and extension of functoriality. A priori, E G (A) appears to be a functor on unbased finite G-sets, but an alternative reformulation is (1.25) E G (A) = P G (A + ) which exhibits E G as a functor on based finite G-sets A +. Here P G is the monad in the category of based G-categories whose algebras are the same as the P G -algebras. Thus P G (A + ) is the free P G -algebra (= genuine permutative G-category) generated by A +, with unit given by the disjoint trivial base category added to A. We need to be more precise about this identification and extended functoriality. Definition Define Λ to be the category of finite based sets n and injections. Formally, P G (A + ) is the categorical tensor product P G Λ A +, where A + sends n to A n +. We make this concrete. Since P G (0) =, there is a degeneracy G-functor σ i : P G(n) P G (n 1) associated to the ordered inclusion σ i : n 1 n that misses i. As in [26, 2.3], if γ is the structural map of the operad and ν P G (n), then σ i (ν) = γ(ν; id i 1,, id n i ). If a i =, then (ν; a 1,, a n ) must be identified with (σi (ν); a 1,, â i,, a n ), where â i means delete a i. Any injection σ : m n, not necessarily ordered, is a composite of such σ i and a unique permutation ρ Σ m. This determines σ : P G (n) P G (m), making P G a contravariant functor on Λ. Define σ : A m +

11 MODELS OF G-SPECTRA AS PRESHEAVES OF SPECTRA 11 A n + by first applying ρ and then inserting the basepoint in the jth slot when j is not in the image of σ, making A + a covariant functor on Λ. Concretely, (1.27) P G (A + ) = n 0(P G (n) Σn A n +)/( ) where is given by (σ µ; a) (µ; σ a) for µ P G (n) and a A m +. Definition For a based G-map f : A + B +, define a functor f! : E G (A) E G (B) by taking the disjoint union over n of the functors id Σn f n. This only uses (1.22) when f 1 ( ) =. 6 In general, however, the specification of f! depends on the functoriality of P on based maps of (1.25) and thus on the basepoint identifications of (1.27). In particular, If i: A B is an inclusion of unbased finite G-sets, define an associated retraction r : B + A + of based finite G-sets by setting ri(a) = a and r(b) = if b / im(a). Then define 7 i = r! : E G (B) E G (A). By Remark 2.21 below, we may think of i as the dual of i. The following definition gives the G-category analogue of Definition 1.6. It specifies a G-category (almost) enriched in permutative G-categories. Definition We define a G-bicategory E G with a permutative G-category of hom objects for each pair of objects as follows. The 0-cells of E G are the finite G-sets A, which may be thought of as the G-categories E G (A). The permutative G-category E G (A, B) of 1-cells and 2-cells A B is E G (B A), as defined in Definition The composition : E G (B, C) E G (A, B) E G (A, C) is given by the following composite. Its first map ω is a pairing of free P G -algebras that will be made precise in Definition Its second and third maps are specializations of the contravariant functoriality of E G on inclusions and its covariant functoriality on surjections, as is made precise in Definition E G (C B) E G (B A) ω E G (C B B A) E G (C A). π! (id id) E G (C B A) This composition is strictly associative. With A = (n, α), E G (A, A) has a pseudounit 1 cell (1.30) A = (α; A ) E G (A A) = P G (n) Σn (A A) n where A = ( (1, 1),, (n, n) ) (A A) n. It is a strict right unit, but it is not a strict left unit (see Remark 1.34 below). 6 With the intuitive version of EG, f! : E G (A) E G (B) is then just the pushforward functor obtained by composing maps over A with f. 7 With the intuitive version of EG, i : E G (B) E G (A) is just the functor obtained by using i to pull back maps over B to maps over A.

12 12 BERTRAND GUILLOU AND J.P. MAY To rectify to obtain a strict unit, we need whiskered G-categories E G analogous to the whiskered categories GE, and we define them in 5. They are defined in such a way that Theorem 1.23 has the following corollary by direct comparison of definitions. Corollary The G-fixed category (E G )G enriched in permutative categories is isomorphic to the category GE enriched in permutative categories. We shall place the following ad hoc definition of the pairing ω required in Definition 1.29 in a general multicategorical context in [15]. We first comment on its domain; compare Remark 1.9. Remark We can define the smash product of based G-categories in the same way as the smash product of based G-spaces (see [6, Lemma 4.20]). We are most interested in examples of the form A + and B + for unbased G-categories A and B, and then A + B + can be identified with (A B) +. In particular, is isomorphic to ( P G (m) Σm A m ) + ( P G (n) Σn B n ) + m 1 n 1 ( m 1,n 1 P G (m) P G (n) Σm Σ n A m B n ) +. We do not claim that this is a P G -category, but the equivariant infinite loop space machine [15] nevertheless constructs from it the smash product of the G-spectra constructed from E G (A) and E G (B). Definition The homomorphism : Σ m Σ n Σ mn defined using lexicographic ordering in Definition 1.3 is the object function of a functor EΣ m EΣ n EΣ mn. Applying the functor Cat G (EG, ), we obtain pairings : P G (m) P G (n) P G (mn); on objects of EG, (µ ν)(g) = µ(g) ν(g). For G-sets A and B, we have the injection : A m B n (A B) mn that sends (a 1,, a m ) (b 1,, b n ) to the set of pairs (a i, b j ), ordered lexicographically. Combining, there result functors ω m,n : (P G (m) Σm A m ) (P G (n) Σn B n ) P G (mn) Σmn (A B) mn, ω m,n ( (µ, a), (ν, b) ) = (µ ν, a b). Distributing products over disjoint unions, these specify pairings of G-categories ω : E G (A) E G (B) E G (A B). Remark The associativity of the composition defined in Definition 1.29 is an easy diagram chase, starting from the associativity of the pairing on P G. We illustrate how Definition 1.28 works by considering composites with the pseudo-unit objects A. Let E be a 1-cell in E G (A, B) and choose an object in the orbit E. (µ; (b 1, a 1 ),, (b m, a m )) P G (m) Σm (B A) m

13 MODELS OF G-SPECTRA AS PRESHEAVES OF SPECTRA 13 We first prove that E A = E. Take A = (n, α). Then the object ( µ α; ((bi, a i, j, j)) ) P G (mn) Σmn (B A A A)mn is in the orbit ω(e, A ). The ordering of the four-tuples is lexicographic on i and j. The four-tuple (b i, a i, j, j) is in the image of id id if and only if a i = j. The r corresponding to this inclusion maps all other (b i, a i, j, j) to the basepoint. Applying π! we arrive at σ ((b 1, a 1 ),, (b m, a m )) (B A) mn, where σ : m mn is the ordered injection that sends i to λ 1 m,n(i, a i ). Therefore E A = (µ α; σ ((b 1, a 1 ),, (b m, a m ))) = (σ (µ α); (b 1, a 1 ),, (b m, a m )). Since σ reverses the lexicographic ordering used to define µ α, we have the reduction σ (µ α) = µ. Now take B = (p, β) and consider B E. Then the object ( β µ; (k, k, bi, a i )) ) P G (pm) Σpm (B B B A) pm is in the orbit ω( B, E). The ordering of the four-tuples is lexicographic on k and i. The four-tuple (k, k, b i, a i ) is in the image of id id if and only if k = b i. The r corresponding to this inclusion maps all other (k, k, b i, a i ) to the basepoint. Applying π! we arrive at τ ((b 1, a 1 ),, (b m, a m )) (B A) pm, where τ : m pm is the injection that sends i to λ 1 p,m(b i, i). But now the injection τ is not ordered, although it becomes so after composition with some ρ Σ m. We have B E = (β µ, τ ((b 1, a 1 ),, (b m, a m )) = (τ (β µ); (b 1, a 1 ),, (b m, a m )), but τ (β µ) is not equal to µ. We define (1.35) l B,E : B E E to be the 2-cell induced by the (unique) morphism τ (β µ) µ in P G (m). The structure E G is only a bicategory, while E G defined in 5 is a strict 2-category. The inclusion E G E G is a pseudofunctor with unit constraint given by ζ. In [14], the category of P G -algebras is the underlying category of a multicategory Mult(P G ). The composition functors in both E G and E G are examples of bilinear maps in the multicategorical sense The categorical duality maps. Since various specializations are central to our work, we briefly recall how duality works categorically, following [20, III 1] for example. We then define maps of P G -algebras that will lead in 2.3 to the proof that the objects of GA are self-dual. Let V be a closed symmetric monoidal category with product, unit S, and hom objects F (X, Y ); write DX = F (X, S). A pair of objects (X, Y ) in V is a dual pair if there are maps η : S X Y and ε: Y X S such that the composites X = S X η id X Y X id ε X S = X Y = Y S id η Y X Y ε id S Y = Y

14 14 BERTRAND GUILLOU AND J.P. MAY are identity maps. For any such pair, the adjoint ε: Y DX of ε is an isomorphism. When (X, Y ) and (X, Y ) are dual pairs, the dual of a map f : X X is the composite (1.36) Y = Y S G id η Y X Y id f id Y X Y ε id S G Y = Y. For any pair of objects X and Y, we have a natural map (1.37) ζ : Y DX = Y F (X, S) F (X, Y ) in V, namely the adjoint of id ε: Y DX X Y S = Y, where ε is the evident evaluation map. The map ζ is an isomorphism when either X or Y is dualizable [20, III.1.3]. When X is self-dual and Y is arbitrary, we have the composite isomorphism (1.38) δ = ζ (id ε): Y X Y DX F (X, Y ). This map in various categories will play an important role in our work. There are two maps of P G -algebras that are central to duality and therefore to everything we do. Let S 0 = {, 1}, where is the basepoint and 1 is not. We think of S 0 as 1 +, where 1 is the one-point G-set. In line with this convention, we also think of 1 as a trivial category with object 1. Remember that E G (A) = P G (A + ) is the free P G -algebra generated by A +, where we view finite G-sets as categories with only identity morphisms. We have already seen the first map implicitly. Definition For a finite G-set A = (n, α), define based G-maps ε: (A A) + S 0, r : (A A) + A + and π : A + S 0 by r(a, b) = if a b and r(a, a) = a, π(a) = 1, and ε = π r, so that ε(a, b) = if a b and ε(a, a) = 1. Note that r = id and that ε is just an example of a Kronecker δ-function. We agree to again write ε for the induced map of P G - algebras ε = E G ε: E G (A A) E G (1). Definition For a finite G-set A = (n, α), regard the object A E G (A A) as the map of G-categories i A : 1 E G (A A) that sends the object 1 of the trivial category to the object A. By freeness, there results a map of P G -algebras η : E G (1) E G (A A). Explicitly, 8 η is the disjoint union over m of the maps given by P G (m) Σm 1 m P G (mn) Σmn (A A) mn η(µ, 1 m ) = ( µ α; ( A ) m). The following categorical observation will lead to our proof in 2.3 that the G-spectra Σ G (A +) are self-dual. Since care of basepoints is crucial, we use the alternative notation P G (A + ). Remember that (A A) + can be identified with A + A +. We identify 1 + A + and A with A + at the bottom center of our diagrams. 8 This uses that γ(µ; α n ) = µ α, where γ : P G (m) P G (n) m P G (mn), as explained in [14, 1].

15 MODELS OF G-SPECTRA AS PRESHEAVES OF SPECTRA 15 Proposition In the diagrams below, square (1) commutes up to isomorphism, and the other three squares commute on the nose. P G (A + A + ) P G (A + ) η id P G (1 + ) P G (A + ) P G (A + ) P G (A + A + ) id η P G (A + ) P G (1 + ) ω P G (A + A + A + ) (1) (2) ω P G (id ε) ω P G (A + ) P G (A + A + ) id ε P G (A + ) P G (A + ) P G (1 + ) ω ω P G (A + A + A + ) ω P G (ε id) ω P G (A + A + ) P G (A + ) ε id P G (A + ) P G (1 + ) P G (A + ) ω Proof. In the right vertical arrows, ε means P G (ε). Both right squares are naturality diagrams, so it remains to consider the squares on the left. The difference between squares (1) and (2) is closely analogous to the difference between left and right composition with A explained in Remark Let A = (n, α) and let (µ, 1 m ) P(m) Σm 1 m and (ν, a) P(q) Σq A q. We consider square (2) first, paying close attention to the order in which variables appear. By Definitions 1.33 and 1.40, and ω ( (ν, a), (µ, 1 m ) ) = (ν µ, a 1 m ) P(qm) A qm ω (id η) ( (ν, a), (µ, 1 m ) ) = ( ν µ α; a ( A ) m ) P G (qmn) Σqmn (A 3 ) qmn. Identifying qm with q m lexicographically, the (k, i)th coordinate of a 1 m is a k. Identifying qmn with q m n lexicographically, the (k, j, i)th coordinate of a ( A ) m is (a k, i, i). By Definition 1.39, ε id sends this coordinate to the basepoint unless a k = i, when it sends it to i. Noticing the agreement of lexicographic orderings, we see as in Remark 1.34 that the injection σ : qm qmn such that σ (a 1 m ) = (ε id) (a ( A ) m ) is ordered and satisfies σ (ν µ α) = ν µ. Now consider square (1). By Definitions 1.33 and 1.40, and ω ( (µ, 1 m ), (ν, a) ) = ( µ ν, 1 m a ) P(mq) Σmq A mq ω (η id) ( (µ, 1 m ), (ν, a) ) = ( γ(µ; α n ) ν; ( A ) m a ) P G (mnq) Σmnq (A 3 ) mnq. Identifying mq with m q lexicographically, the (i, k)th coordinate of 1 m a is a k. Identifying mnq with m n q lexicographically, the (i, j, k)th coordinate of ( A ) m a is (j, j, a k ). By Definition 1.39, id ε sends this coordinate to the basepoint unless j = a k, when it sends it to j. Here the injection τ : mq mnq such that τ(1 m a) = (id ε) (( A ) m a) is not ordered, although it becomes so after composition with some ρ Σ mq, and τ (µ α ν) is not equal to µ ν in P G (mq). As in Remark 1.34, there is a unique 2-cell, necessarily an isomorphism, ϑ: (µ ν) = τ (µ α ν)

16 16 BERTRAND GUILLOU AND J.P. MAY in P G (mq). As the input varies, the 2-cells (ϑ, id): ( µ ν; 1 m a) = ( τ (µ α ν), 1 m a ) specify the 2-natural isomorphism in the square (1). 2. The proof of the main theorem 2.1. The equivariant approach to Theorem As we explain in [15], following [9], equivariant infinite loop space theory associates an orthogonal G-spectrum K G C G to a genuine permutative (or more generally gemuine symmetric monoidal) G-category C G. The 0th space of K G C G is an equivariant group completion of the classifying G-space BC G. 9 The category GS of orthogonal G-spectra is the G- fixed category of a G-category S G of G-spectra and non-equivariant maps with the same objects as S G and with G acting by conjugation on morphisms. Applying the functor K G to E G, we obtain the following equivariant analogue of Definition Definition 2.1. We define a G-spectral category, or S G -category, A G. Its objects are the finite G-sets A, which may be viewed as the G-spectra K G E G (A). Its morphism G-spectra A G (A, B) are the K G E G (B A). Its unit G-maps S G A G (A, A) are induced by the points I A GE (A, A) (see 5) and its composition G-maps are induced by composition in E G. A G (B, C) A G (A, B) A G (A, C) Again, as written, the definition makes little sense: to make the word induced meaningful requires properties of the equivariant infinite loop space machine K G that we will spell out in 2.2. This depends on having a functor that takes pairings (alias bilinear maps) of free P G -algebras to pairings of G-spectra. The equivariant and non-equivariant infinite loop space functors are related by the following result. Theorem 2.2 ([9]). There is a natural equivalence of spectra ι: K(GC ) (K G C G ) G for permutative G-categories C G with G-fixed permutative categories GC. In view of Corollary 1.31, there results an equivalence of S -categories GA (A G ) G. The proof of Theorem 1.13 goes as follows. We start with the following specialization of a general result about stable model categories; it is discussed in 4.1. The essential point is that the collection {Σ G A +} is a set of generators for HoGS. 9 The papers from around 1990, such as [4, 37] are not adequate, in part because genuine permutative G-categories were not explicitly defined and the group completion property was not worked out rigorously, but more substantially because a symmetric monoidal category of G-spectra had not yet been discovered. A key feature of the version of the Segal machine [11] used in our proofs is that it is given by a symmetric monoidal functor, a claim that would not have made sense in 1990.

17 MODELS OF G-SPECTRA AS PRESHEAVES OF SPECTRA 17 Theorem 2.3. Let GD be the full S -subcategory of GS whose objects are fibrant approximations of the suspension G-spectra Σ G (A +), where A runs through the finite G-sets. Then there is an enriched Quillen adjunction and it is a Quillen equivalence. Pre(GD, S ) T GS, U Here GD is isomorphic to (D G ) G, where D G is a full S G -subcategory D G of S G. Theorem 2.4 (Equivariant version of the main theorem). There is a zigzag of weak equivalences connecting the S G -categories A G and D G. A weak equivalence between S G -categories with the same object sets is just an S G -functor that induces weak equivalences on morphism G-spectra. 10 On passage to G-fixed categories, this equivariant zigzag induces a zigzag of weak S -equivalences connecting the S -categories GA and GD. In turn, by [8, 2.4], this zigzag induces a zigzag of Quillen equivalences between Pre(GA, S ) and Pre(GD, S ). Since Pre(GD, S ) is Quillen equivalent to GS, it follows that Theorem 2.4 implies Theorem Remark 2.5. The functor U sends G/H to F G (Σ G G/H +, X) G = X H. Keeping that fact in mind shows why Corollary 1.14 follows from the proof of Theorem To understand GS as an S -category, we must first understand S G as an S G - category. That is, to understand the G-fixed spectra F G (X, Y ) G, we must first understand the function G-spectra F G (X, Y ). Using infinite loop space theory to model function spectra implicitly raises a conceptual issue: there is no known infinite loop space machine that knows about function spectra. That is, given input data X and Y (permutative G-categories, E -G-spaces, Γ-G-spaces, etc) for an infinite loop space machine K G, we do not know what input data will have as output the function G-spectra F G (K G X, K G Y ). The problem does not even make sense as just stated because the output G-spectra K G X are always connective, whereas F G (K G X, K G Y ) is generally not. The most that one could hope for in general is to detect the connective cover of F (K G X, K G Y ). In our case, the relevant function G-spectra are connective since the suspension G-spectra Σ G (A +) are self-dual, as we shall reprove in Results from equivariant infinite loop space theory. The proof of Theorem 2.4 is the heart of this paper, and of course it depends on equivariant infinite loop space theory and in particular on the relationship between the G-spectra A G (A) = K G E G (A) and the suspension G-spectra Σ G (A +). We collect the results that we need from [15] in this section. We warn the skeptical reader that the results of this paper depend fundamentally on Theorems 2.6 and 2.8. However, the proofs of those results require work far afield from the applications in this paper. In fact, Theorem 2.4 is an application of a categorical version of the equivariant Barratt-Priddy-Quillen (BPQ) theorem for the identification of suspension G- spectra. 11 We state the theorem in full generality before restricting attention to finite G-sets. We shall find use for the full generality in A more general definition is given in [8, 2.3]. 11 For A =, Carlsson [3, p.6] mentions a space level version of the BPQ theorem. Shimakawa [37, p. 242] states and gives a sketch proof of a G-spectrum level version.

18 18 BERTRAND GUILLOU AND J.P. MAY Recall from Remark 1.24 that E G (A) can be identified with the category P G (A + ), where P G is the free P G -category functor on based G-categories. The functor P G applies equally well to based topological G-categories. 12 We view a based G-space X as a topological G-category that is discrete in the categorical sense: its morphism and object G-spaces are both X, and its source, target, identity, and composition maps are all the identity map of X. Thus we have the topological P G -category P G (X). The geometric realization of its nerve is the free E G-space generated by X. Henceforward, we use the term stable equivalence, rather than weak equivalence, for the weak equivalences in our model categories of spectra and G-spectra. We are only interested in the following results for based G-spaces of the form X +, but we state the slightly more general version that holds for all based G-spaces. It holds by [9, Theorems 6.1 and 6.2]. Theorem 2.6 (Equivariant Barratt-Priddy-Quillen Theorem). For based G-spaces X, there is a natural stable equivalence α: Σ G X K G P G (X). Of course, the naturality statement says that the following diagram commutes for a map f : X Y of based G-spaces. (2.7) Σ G X α K G P G (X) Σ G f Σ G Y K α G P G (Y ) K G P G (f) In order to produce our spectral category A G, it is essential that we have a machine with good multiplicative properties. The following result, which is proven in [15], gives far more than we need. As explained there, we have a multicategory Mult st (P G ) of strict P G -algebras; it is a submulticategory of a multicategory Mult(P G ) of P G -pseudoalgebras, but its multilinear maps require P G - pseudomaps despite the restriction to strict P G -algebras as objects. We also have the multicategory Mult(S G ) associated to the symmetric monoidal category of orthogonal G-spectra under the smash product. Theorem 2.8. [15] K G extends to a multifunctor K G : Mult(P G ) Mult(S G ). Remark 2.9. At one place in the duality proof of 2.3 below, we use from [15] that K G converts 2-cells, such as ϑ in Proposition 1.41, to homotopies between maps of G-spectra. We have a more down to earth corollary that relates α to smash products and, together with accompanying associativity and unit conditions, gives all that we really need. Observe that the pairing ω of Definition 1.33 generalizes to give a natural pairing ω : P G (X + ) P G (Y + ) P G (X + Y + ) for unbased G-spaces X and Y. 12 We understand a topological G-category to mean an internal category in the category of G-spaces.